Optimal wavelet filter banks for regularized restoration of noisy images

Regularized image restoration methods efficiently handle the ill-posed prob lem of image restoration. Nevertheless, the issue of selecting the regulari zation parameter as well as the smoothing filter still constitutes an open research topic. A model of regularized image restoration is introduced and analyzed in this paper. The proposed model assumes that wavelet filter bank s replace the smoothing filter of conventional regularized restoration. Fil ter factorizations for the optimal design of wavelet filter banks using the generalized-cross-validation (GCV) criterion are presented, and novel expr essions of the influence matrix, which is used to calculate the GCV error, are derived. The error of the GCV method is expressed in terms of the modul ation matrix of the filter bank and the modulation vector of the degradatio n filter. The expressions are given in general form for optimal wavelet fil ter bank design upon arbitrary sampling lattices. The numerical examples of image restoration using the proposed method that are presented indicate si gnificant signal-to-noise ratio improvement, Delta(SNR), compared to image restoration methods that employ the Laplacian as the smoothing filter.